Question

If $b+c=3a,$ then the value of $\cot \left(\frac{B}{2}\right)\cot \left(\frac{C}{2}\right)$ is equal to

• A. $1$
• B. $2$
• C. $\sqrt{3}$
• D. $\sqrt{2}$

Answer 1

The correct option is B $2$

$\cot \left(\frac{B}{2}\right)\cot \left(\frac{C}{2}\right)$
$=\sqrt{\frac{s(s-b)}{(s-a)(s-c)}}\sqrt{s(s-c)}(s-a)(s-b)$
$=\frac{s}{s-a}$ (on simplification)
$=\frac{2s}{2(s-a)}$ (multiply and divide by $2$)
$=\frac{2s}{2s-2a}$
We know that $a+b+c=2s$
$=\frac{a+b+c}{a+b+c-2a}$
$=\frac{a+b+c}{b+c-a}$
Since $b+c=3a$(given)
$=\frac{a+3a}{3a-a}=\frac{4a}{2a}=2$